R.  Feistel  et  al.:  Oceanographic  application  and  numerical  implementation  of  TEOS-IO:  Part  1

647

www.ocean-sci.net/6/633/2010/

Ocean  Sci.,  6,  633-677,  2010

and  the  potential  density  from  Eq.  (S14.1),  pe,  library  function ­
  air_potdensity_si,

-l

Pg

dh AW (A,  i],P r )\
SPr  ) A ' V

(4.45)

In  terms  of  the  Primary  Standard  functions  and  their  independent ­
  variables  (Sect.  2),  Eq.  (5.1)  is  expressed  by  the  system
^Jr 1  (  7'  ^  W  \  ^

/ F (i> w )  +  P7„ F (r,„'*')
=  / F (r,p'j+p'7J(r,p'j

Evidently,  for  any  fixed  reference  pressure,  P r ,  the  value  of
/z AV  (A,r),  P r )  and  its  partial  derivatives,  as  well  as  any  other
arbitrary  function  depending  on  this  triple  of  variables,  remain ­
  unchanged  during  isentropic  processes  (rj  =  const)  at
constant  specific  humidity  (A  =  const).
Physically  reasonable  values  of  the  entropy  to  be  used
as  an  independent  variable  of  the  enthalpy  are  restricted  to
ranges  depending  on  humidity  and  pressure,  between  the  particular ­
  limits  given  by  dry  and  saturated  air,  see  Sect.  5.8.

5  Level  4:  Phase  equilibria  and  composite  systems
Equilibrium  properties  at  phase  transition  boundaries  or  of
coexisting  phases  are  often  characterized  by  drastic  spatial
or  temporal  changes,  and  large  values  of  latent  heat  exchange
or  volume  expansion,  e.g.  if  seawater  freezes  or  evaporates.
Such  multi-phase  and  multi-component  properties  are  available ­
  from  combinations  of  the  thermodynamic  potentials  if
they  are  consistently  adjusted  to  reference  state  conditions
which  fix  the  absolute  energies  and  entropies  of  the  substances ­
  involved  (Feistel  et  al.,  2008).  Gibbs  functions  can
be  constructed  for  composite  systems  such  as  sea  ice  (Feistel
and  Hagen,  1998,  Sect.  5.4)  that  contain  two  stable  phases
(e.g.  ice  and  seawater).  When  the  temperature,  the  volume
or  the  pressure  of  a  composite  system  is  changed,  mass  is
transferred  from  one  phase  to  the  other;  for  example  if  sea
water  freezes  or  evaporates,  brine  salinity  or  vapour  pressure
adjust  to  the  new  conditions  imposed  and  the  heat  capacity
or  the  thermal  expansion  of  the  whole  system  exhibits  very
large  changes  resulting  from  the  changes  in  latent  heat  contributions. ­
  By  utilizing  mutually  consistent  potential  functions,
rigorous  mathematical  formulae  can  be  determined  for  the
numerical  calculation  of  latent  properties  depending  on  the
particular  conditions  such  as  isobaric,  isochoric  or  isentropic
processes.
5.1  Equilibrium  liquid  water-vapour:  saturation
The  saturation  point  of  pure  water  is  usually  computed  at
a  given  temperature  T,  providing  the  vapour  pressure  P  —
P vap (r),  or  at  a  given  pressure  P  providing  the  boiling  temperature ­
  T  —  r boll (P).  The  defining  condition  is  equality  of
the  chemical  potentials  of  liquid  and  vapour,  which  equal  the
Gibbs  functions  in  the  case  of  pure  phases,
g w (P,P)  =g w (T,P).  (5.1)

(p W f  fp( T ’P W )  = P  (5-3)
(p v ) 2 /j(r,p v )=P  (5.4)
which  exploits  the  relations  (Eqs.  S2.6  and  S2.ll)  to  avoid
stacked  numerical  iterations.  Eq.  (5.2)  is  equivalent  to
Eq.  (5.1)  and  is  also  known  as  the  “Maxwell  condition”  in
p w
the  form  /  |Vj—^-Jdp=0.  Equations  (5.2)-(5.4)  provide
p v
three  equations  for  the  four  unknowns  T,  P,  p v  and  p w .
Any  one  of  these  quantities  can  be  specified  independently
to  complete  the  system  and  permit  the  numerical  solution  as
discussed  in  Appendix  A3.
Once  the  values  of  T,  P,  p v  and  p w  are  computed  from
the  iteration  of  Eqs.  (5.2)-(5.4)  at  the  specified  saturation
condition,  various  equilibrium  properties  can  be  determined
from  the  formulae  given  in  Table  S15.
5.2  Equilibrium  water-ice:  melting  and  freezing
The  melting  pressure  of  ice  is  usually  computed  at  a  given
temperature  T,  giving  P melt (r).  Similarly,  the  freezing  temperature ­
  of  water  is  normally  determined  at  a  given  pressure
P,  giving  P frz (P),  which  also  gives  the  melting  temperature
of  ice.  In  either  case,  the  defining  condition  is  equality  of  the
chemical  potentials  of  liquid  water  and  ice,
g w (T,P)  =g m (T,P).  (5.5)
In  terms  of  the  Primary  Standard  functions  and  their  independent ­
  variables  (Sect.  2),  Eq.  (5.5)  is  represented  as  the  system
f  (P,  p w )  +  p w /J  (p,  p w )  =  g lh (T,  P)  (5.6)
(p W f  fp( T 'P W )  = P ’  (5 ' 7)
which  exploits  the  relations  (Eqs.  S2.6  and  ES2.11)  to  avoid
stacked  numerical  iterations.  Equations  (5.6)  and  (5.7)  supply ­
  two  equations  for  the  three  unknowns  P,  P  and  p w .
Specifying  any  one  of  these  quantities  completes  the  determination ­
  of  the  system  which  can  then  be  solved  as  discussed
in  Appendix  A4.
Once  the  values  of  P,  P  and  p w  are  computed  from  the
iteration  of  Eqs.  (5.6),  (5.7)  at  the  specified  melting  condition, ­
  various  equilibrium  properties  can  be  determined  from
the  formulae  given  in  Table  S16.